Vibrations Of Rods Research Articles

Dynamic stiffness method for exact longitudinal free vibration of rods and trusses using simple and advanced theories

Closed-form dynamic stiffness (DS) formulations coupled with an efficient eigen-solution technique are proposed for exact longitudinal free vibration analyses of rods and trusses by using classical, Rayleigh-Love, Rayleigh-Bishop and Mindlin–Hermann theories. First, the exact general solutions of the governing differential equations of the four rod theories are developed. Then the solutions are substituted into the generalized displacement and force boundary conditions (BCs), leading to the elemental DS matrices utilising symbolic computation. As an accurate and efficient modal solution technique, the Wittrick-Williams (WW) algorithm is applied. The J0 count for the WW algorithm has been resolved for all four types of DS elements with explicit analytical expressions. The method is verified against some existing exact results for rods subjected to specific BCs. Comparisons of the natural frequencies and mode shapes for different theories and slenderness ratios are also made. Finally, benchmark solutions are provided for individual rods subject to different BCs, a stepped rod and a truss. This research provides an exact and highly efficient modal analysis tool for rods and trusses within the whole frequency range, which is suitable for parametric studies, optimization design, inverse problem analysis, and important for statistical energy analysis.

A PROBLEM WITH NONLOCAL CONDITION FOR ONE-DIMENSIONAL HYPERBOLIC EQUATION

In this paper, we study the problem with a dynamic nonlocal condition for the one-dimensional hyperbolic equation, which occurs in the study of rod vibrations. This problem may be used as a mathematical model oflongitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. Conditions have been obtained for input data, providing unambiguous resolution of the task, proof of the existence and singularity of the problem in the space of Sobolev. The proof is based on the a priori estimates obtained in this paper, Galerkins procedure and the properties of the Sobolev spaces.

Study on the bending vibration of bimorph rectangular transducer based on type 2-2 piezoelectric composites

In order to improve the acoustic radiation capability of piezoelectric ultrasonic transducer in gas, a bimorph rectangular transducer based on type 2-2 piezoelectric composites is proposed in this paper, which is composed of two piezoelectric composite rectangular slices with thickness polarization. First, the bending vibration of the rectangular transducer is divided into two bending vibrations of the thin rods in the x and y directions using the equivalent elastic method. When the mechanical coupling coefficient is introduced and combined with the anisotropic piezoelectric equations, the resonance frequency equations of the thickness-polarized transducer are derived with simply supported boundary conditions. Then the bending vibration of the transducer is numerically simulated. The results show that the resonance frequency obtained by the analytical method is in good agreement with the numerical simulation results. Finally, the influence of geometrical size and two-phase volume ratio on the bending vibration performance of the transducer is analyzed. Compared with the piezoelectric bimorph transducer, the piezoelectric composite bimorph transducer has a larger bending vibration displacement amplitude, a larger transmitting voltage response, a lower resonant frequency and better acoustic radiation capability.

AIMED CONTROL OF THE FREQUENCY SPECTRUM OF EIGENVIBRATIONS OF ELASTIC PLATES WITH A FINITE NUMBER OF DEGREES OF FREEDOM OF MASSES BY SUPERIMPOSING ADDITIONAL CONSTRAINTS

As is known, for some elastic systems with a finite number of degrees of freedom of masses, for which thedirections of motion of the masses are parallel and lie in the same plane, methods have been developed for creatingadditional constraints that purposefully change the spectrum of natural frequencies. In particular, theory and algorithm forthe formation of aimed additional constraints have been developed for the rods, the introduction of each of which doesnot change any of the modes of natural vibrations, but only increases the value of only one frequency, without changingthe values of the remaining frequencies. The distinctive paper is devoted to the method of forming a matrix of additionalstiffness coefficients corresponding to such aimed constraint in the problem of natural vibrations of rods. This method canalso be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which thedirections of motion of the masses are parallel, but not lie in the same plane. In particular, such systems include plates.However, the algorithms for the formation of aimed additional constraints, developed for rods and based on the propertiesof rope polygons, cannot be used without significant changes in a similar problem for plates. The method for the formationof design constraint schemes that purposefully change the spectrum of frequencies of natural vibrations of elastic plateswith a finite number of degrees of freedom of masses, will be considered in the next work.

НЕЛИНЕЙНЫЕ ПОПЕРЕЧНЫЕ КОЛЕБАНИЯ СОСТАВНЫХ СТЕРЖНЕЙ ПРИ ДЕЙСТВИИ СТАТИЧЕСКИ ПРИЛОЖЕННОЙ ПОПЕРЕЧНОЙ НАГРУЗКИ

Nonlinear transverse vibrations of composite rods pre-loaded with lagging arranged symmetrically on both sides of the axis of the composite rod under the action of a statically applied transverse load are investigat-ed. The cases of attaching the lagging only to the ends of the composite rod, as well as when the laggings are continuously attached to the composite rod along its entire length, are considered. The results of the study of nonlinear transverse vibrations of composite rods under the action of a statically applied transverse load are presented. When conducting studies of transverse vibrations of composite rods, solutions of differential equations of vibration of prestressed through beams and stiffening cores of high-rise buildings are obtained. The obtained differential equations of vibration of composite rods allow us to determine the dynamic char-acteristics of prestressed through beams under various linear and boundary conditions. A method for com-posing differential equations of free and forced oscillations of prestressed through beams and stiffening cores of high-rise buildings and solving differential equations under various linear and nonlinear boundary conditions is developed.
 Expressions are given for determining the longitudinal forces and torques at the ends of the rod at any lo-cation of the lagging from the axis and at any different stiffness of the lagging

Nonlinear stochastic vibration of a variable cross-section rod with a fractional derivative element

The article deals with the problem of computing efficiently the nonlinear response of a rod involving a fractional constitutive model, and exposed to random excitation. The constitutive model is a three-parameter model comprising an instant elasticity modulus, a prolonged elasticity modulus, and a relaxation parameter. The nonlinear term is a linear-plus-cubic force of the Winkler kind. The resulting nonlinear fractional partial differential equation governing the rod displacement has no known exact solution. Thus, the article proposes an approximate analytical solution by relying on the statistical linearization technique. Further, it develops a Boundary Element Method (BEM)-based approach to estimate numerically the rod response statistics. The statistical linearization solution is obtained by representing the rod displacement as the superposition of linear modes of vibration having time-dependent coefficients. In this context, it is shown that the equation governing the time variation of the mode coefficients is a nonlinear fractional ordinary differential equation, whose solution is computed by a surrogate linear system identified by minimizing the response error between the linear system and the nonlinear one in a mean square sense. Relevant Monte Carlo studies pertaining to rods with fixed-fixed, and fixed-free ends show that the proposed analytical solution is in a good agreement with data obtained by the numerical (BEM) approach.

Visualization experimental study on axial flow-induced vibration of slender cantilever rod for reactor

Experimental visualization is conducted on axial flow-induced vibration (FIV) of a slender cantilever rod in a narrow annular flow channel with a nonintrusive measurement. A real cobalt rodused in PWR was studied. A high-speed camera was employed to capture the vibration displacement response and deformation in a narrow channel. The test rod shows random low-frequency vibration before touching the tube wall, and the vibration amplitude of the test rod continuously increasing with Reynolds number. When the rod touches the tube wall, the vibration amplitude decreases sharply. When Reynolds number is around 1820, the rod shows first-order vibration divergence, which has a large deformation with a tiny vibration amplitude. This work provides a novel and direct approach for measuring the displacement response and deformation of axial FIV in the narrow channel. Moreover, current experiment data can be also used to verify the numerical simulation for axial FIV of elastic slender cantilever rod in water flow.

FREE TORSIONAL VIBRATIONS OF A ROUND ELASTIC CONE-SHAPED ROD

t. The article deals with the problem of torsional vibrations of circular elastic conical rods in the system of cylindrical coordinates. It is believed that vibrations of the rod are caused by external dynamic loads applied to its surface. The cross-sectional radius is taken as a linear function of the longitudinal coordinates. On the basis of the equation of motion of the theory of elasticity for torsional vibrations, the equation of torsional vibrations of a conical rod with a circular cross-section is derived using the Fouret and Laplace integral substitutions. On the basis of the obtained equation, the problem of harmonic torsional vibrations of a rod is solved. The frequency equation and the relationship between frequency and number of waves are given. Based on the obtained numerical results, appropriate conclusions were drawn

TORSIONAL VIBRATIONS OF A CIRCULAR ELASTIC ROD TAKING INTO ACOUNT PHYSICAL NONLINEARITY

. In this work, on the basis of the equations of torsional vibrations of a circular rod derived by the author, taking into account physical nonlinearity, the problem of torsional vibrations of a rod is numerically solved. A comparative analysis of the results obtained for nonlinear and linear cases is carried out. To solve the problem, a numerical finite difference method is applied. The approximation of the oscillation equation and boundary conditions leads to a system of algebraic equations, the solution of which is not mathematically difficult. On the basis of the obtained numerical data of the problem, graphs of the dependences of the torsional displacement and stresses on time were constructed. The main conclusions made on the basis of the constructed graphs of displacement and stresses are presented.

Моделирование продольно-крутильного преобразователя ультразвукового медицинского инструмента

The article considers small vibrations of a spatial helical rod included in the longitudinal-torsional transducer of an ultrasonic medical instrument. An algorithm for determining the natural frequencies and waveforms of system vibrations by the method of initial parameters is developed. On the basis of this algorithm the real elastic element of the longitudinal-torsional transducer is calculated using the mathematical package MATLAB. The obtained natural frequencies allow ensuring the operation of the ultrasound medical instrument in the resonant mode.

Analysis of the neutron noise induced by fuel assembly vibrations

The investigation of neutron noise is key to several applications in nuclear reactor physics, such as the detection of control rod or assembly vibrations and the diagnostic of coolant speed and void fraction. In this paper we will elucidate some aspects of the noise equations in the Fourier domain, for the case of periodic fuel rod vibrations with frequency ω0 in a small symmetrical system in which the perturbation is centrally located. We will in particular focus on the double frequency effect, i.e., the emergence of a noise component at 2ω0 (possibly stronger than the one at the fundamental frequency ω0). Our analysis will be carried out without truncating the noise source at the first order and in the context of a non-perturbative approach (i.e., without resorting to linearization). For this purpose, we will select a simple benchmark configuration that is amenable to accurate reference solutions obtained by solving the exact time-dependent transport equations. The analysis carried out in this work suggests that the non-perturbative noise equations are mandatory in order to properly discriminate the possible emergence of double frequency effects in neutron noise, especially in view of comparing simulation results to experimental data.

Forced vibrations of size-dependent rods subjected to: impulse, step, and ramp excitations

Forced longitudinal vibration response of nonlocal strain gradient (NLSG) rods is scrutinized and compared with nonlocal (NL), strain gradient (SG), and classical (CL) rods. In this respect, size-dependent kinematics and extended Hamilton’s principle are utilized to derive governing equations of motion. For the first time, forced longitudinal vibratory behavior of non-classical and classical rods is presented including three different types of non-harmonic forcing functions: unit impulse, unit step, and unit ramp. Forcing terms are functions of both temporal and spatial domains. Nonlocal strain gradient and nonlocal rod models take into account the Laplacian of the external forcing term as well. This implies that in models including nonlocal effects (NL, NLSG-h, s, n), forcing term is reduced by the order of its Laplacian with respect to spatial variables. Effect of non-classical parameters over time-domain response of the system is analyzed in details. Results are compared with benchmark for the sake of evaluation. This implies the necessity of adopting proper values of non-classical theory and corresponding parameters through analysis of forced oscillations of small-scaled size-dependent rods. Such external excitations are modeled using spatial Dirac delta function as a concentrated force applying at the critical point of mode shape function. Findings of this research reveal the underlying impact of non-classical theorems along with corresponding non-classical parameters over the prediction of the response of size-dependent rod elements for wide range of applications such as in musical instruments, navigating equipment, gyroscopes, control of robotics, sensors and actuators, resonators, energy harvesters, and vibration control systems.

Longitudinal vibrations of a cylindrical bar from the action of harmonic forces

The longitudinal vibration of a cylindrical rod has been studied in an elastic medium from the action of axial harmonic forces in a state of compression tension. The consideration took into account the damping of vibration of the rod according to the theory of E.S. Sorokin. Taking into account, internal friction made it possible to approach the real state of the rod as a source of wave radiation. The amplitude-frequency characteristic of the structure is built, taking into account changes in the parameters of the environment and the rod. The possibilities of the emergence of the resonance state of the rod are established. At the same time, cases were considered with and without inelastic properties of the rod material. The analysis of the influence of the properties of the environment and the rod on the amplitude of the displacement of the latter is carried out.

Longitudinal vibrations of rods

Elastic structures of buildings and machines, the dynamic behavior mathematical model of which is the problem of longitudinal vibrations of rods, are widespread in modern technology. In this regard, the study of issues related to the longitudinal vibrations of the rods is also an urgent problem currently. In terms of solving such problems, we considered the problem formulation of longitudinal (free and forced) vibration of rods, obtained the spectra of natural frequencies ωn and own forms φn(x) vibration, u(x, t) – the function of cross-sections displacement in the longitudinal direction of the rod has been found.

Нелинейное возбуждение гиперзвуковых колебаний в ферритовой пластине в условиях комбинированного воздействия на двух частотах. Часть 3. Вариация толщины пластины

The task about nonlinear excitation of hypersound vibrations in ferrite plate in conditions of combine influence in two frequencies is investigated. As a most important parameter which is varied it is proposed the relative thickness of plate which is determined as relation of real thickness to the thickness which correspond to elastic resonance on the difference of excitation frequencies. It is established the necessity of choosing of character value of constant field which is determined by enough effective excitation of elastic vibrations. The system of nonlinear equations of motion of magnetization and elastic displacement is described. For solving of this system, the numerical Rounge-Cutta method is applied. The results of this calculation are the time-evolvent of vibrations, dependencies magnetic end elastic vibrations amplitudes and the spectra of vibrations in permanent conditions after end of relaxation processes. It is found the multi-regime character of elastic vibrations which takes place by variation of plate thickness. In the character of development of elastic vibrations in time by the increasing of plate thickness it is found four regimes: regime №1 – regular beatings, regime №2 – established resonance, regime №3 – displacement of center of established vibrations, regime №4 – gigantic oscillations. The intervals of thickness values which are necessary, or realization of these regimes are determined. The properties of each regimes taken separately are investigated. It is found that the regime №1 is realized when the thickness of plate is more less then the thickness of resonance on differential frequency. In this case the elastic vibrations in generally repeats the vibrations of magnetization which are realized as beating between two frequencies of excitation. The regime №2 takes place when the plate thickness is near to resonance on differential frequency. When thickness is corresponds to resonance on differential frequency it is found large raising of resonance character. In the vibrations of elastic displacement, the constant component is discovered. The regime №3 takes place when the plate thickness is exceeded of resonance on several (from two to seven) times. The vibrations of magnetization in this regime are the same as in regimes №1 and №2. The elastic displacement has two components: oscillatory on differential frequency and constant which value by increasing of thickness smoothly is increased. The displacement of center of oscillatory component by thickness is increased has quadratic character. The regime №4 takes place by plate thickness exceeds resonance thickness on the order and more. The vibrations of magnetization maintain the character of beating which are the same as in regimes №1, №2 and №3. The vibrations of elastic displacement are characterized by extremely large amplitude which is more then the amplitude in regime №3 on order and more and has extremely large period which is more then period of differential frequency vibrations on two-three order and more. The amplitude of vibrations and its period by the thickness is increases also increase by linear meaning. The some quality opinions about the nature of observed phenomena are proposed. It is established the specific character of two-frequency excitation in comparison to single-frequency excitation. As the possible task it is proposed the plan of singing the part of solution as dependence of vibration amplitude from plate thickness has quadratic character with necessary appreciation of two-frequency excitation. The mechanical analogy for vibrations of hard rod which is compressed on both ends by approaching forces is proposed. This analogy allows to interpret the displacement of vibrations center and gigantic oscillations regime.

Transverse vibrations of an underground cylindrical structure

The transverse vibrations of an extended cylindrical underground structure under the action of concentrated forces changing according to a harmonic law are considered. The problem of transverse vibrations of a cylindrical rod interacting with an elastic homogeneous medium was solved by the Fourier method. Approximate solutions are obtained using the method of compensating loads. The influence of soil conditions and geometrical dimensions of the structure, the location of active forces on the amplitude-frequency characteristics of the object are analyzed.

Studying free hight-frequency vibrations of an inhomogeneous nanorod based on the nonlocal theory of elasticity

Free high-frequency longitudinal vibrations of an inhomogeneous nanosized rod are studied on the basis of the nonlocal theory of elasticity. The upper part of spectrum with the wavelength comparable to the internal characteristic dimension of a nanorod is examined. An equations in the integral form with the Helmholtz kernel, incorporating both local and nonlocal phases, is used as the constitutive one. The original integro-differential equation is reduced to the forth-order differential equation with variable coefficients, the pair of additional boundary conditions being deduced. UsingWKB-method, a solution of the boundaryvalue problem is constructed in the form of the superposition of a main solution and edge effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model which allows estimating the upper part of spectrum of eigen-frequencies. Considering the nanorod with a variable cross-section area, we revealed a fair convergence of eigen-frequencies found in the framework of two models when the local fraction in the two-phase model vanishes.

Direct Numerical Simulation of Aeroelastic Vibrations of a High-Aspect Rod for Modes Similar to Resonance Modes

Direct Numerical Simulation of Aeroelastic Vibrations of a High-Aspect Rod for Modes Similar to Resonance Modes

NEUTRON NOISE-BASED ANOMALY CLASSIFICATION AND LOCALIZATION USING MACHINE LEARNING

A methodology is proposed in this paper allowing the classification of anomalies and subsequently their possible localization in nuclear reactor cores during operation. The method relies on the monitoring of the neutron noise recorded by in-core neutron detectors located at very few discrete locations throughout the core. In order to unfold from the detectors readings the necessary information, a 3-dimensional Convolutional Neural Network is used, with the training and validation of the network based on simulated data. In the reported work, the approach was also tested on simulated data. The simulations were carried out in the frequency domain using the CORE SIM+ diffusion-based two-group core simulator. The different scenarios correspond to the following cases: a generic “absorber of variable strength”, axially travelling perturbations at the velocity of the coolant flow (due to e.g. fluctuations of the coolant temperature at the inlet of the core), fuel assembly vibrations, control rod vibrations, and core barrel vibrations. In all those cases, various frequencies were considered and, when relevant, different locations of the perturbations and different vibration modes were taken into account. The machine learning approach was able to correctly identify the different scenarios with a maximum error of 0.11%. Moreover, the error in localizing anomalies had a mean squared error of 0.3072 in mesh size, corresponding to less than 4 cm. The proposed methodology was also demonstrated to be insensitive to parasitic noise and will be tested on actual plant data in the near future.

Analyzing the Vibrations of the Critical Components in the Power Generating Unit of 200MW under Kinematic Loadings

An important problem that should be resolved during the power plant operation is related to the reliability control for all modes of operation including extreme external dynamic effects of a natural character, in particular the earthquakes. The vibrations of the elements of power equipment exposed to seismic forces occur due to kinematic perturbations that are transmitted from the ground to the bottom base plate. In this connection, the following requirements are set up to the seismic stability of the power equipment of power plants that include its strength, impermeability and its serviceability when exposed to seismic forces. The loading of the fasteners of the turbine generating unit of 200MW arranged on the base was analyzed in the case of exposure to seismic forces. Calculation methods and the software were used that were developed using the finite element method. The computational model of the dynamic system, in particular the turbine generating unit- bed and base consisted of arbitrary- oriented rods and lumped masses that are rigidly interconnected with each other or these can also be interconnected by linear and nonlinear elastically-damped elements. The rod finite elements with distributed parameters allow us to take into consideration all the types of deformations that occur during rod vibrations. The computational method allows us to take into consideration the structural peculiarities of the system and the peculiarities of its behavior during possible earthquakes. The kinematic perturbation that simulates the seismic action transmitted from the ground to the bottom bed plate is assigned by the accelerogram that includes acceleration in three mutually perpendicular directions. The computations were done for the synthesized seismogram of the earthquake of wind force 7 that results in the broadest spectrum of the loading in comparison with other real seismograms. The analysis of the computational data obtained for the seismic stability of the turbine generating unit of 200MW showed that the displacement of rotors towards the turbine generating unit axis relatively the cylinder bodies and the displacement of the upper bed plate relatively the bottom bed plate is lower than permissible values in the case of the earthquake of wind force 7.The keyed joints are the most loaded system assemblies though the strains that arise in them during the kinematic perturbations in question also fail to attain permissible values. The obtained research data are of practical importance and can be used for the assessment of the seismic stability of turbine generating units of a different power.